Covariate Effects Shared Across Recurrences

Why Share a Covariate Effect?

A recurrent-event analysis can distinguish the first event from the second, third, and later events. In a Prentice–Williams–Peterson (PWP) model, for example, the risk set for the $k$th event includes only individuals who have already experienced $k-1$ events. This conditional structure is useful, but it raises a modeling choice: should a covariate have the same relative effect at every event order, or should its effect be allowed to vary?

A shared effect is a substantive restriction, not merely a convenient way to shorten a coefficient table. It says that a one-unit covariate contrast multiplies the conditional event hazard by the same factor at each modeled event order, under the model’s other assumptions. The baseline hazards may still differ by order, and the risk sets themselves may differ.

Model and Meaning of a Shared Effect

Definition: Let $T_{ik}$ be the time to the $k$th event for individual $i$, and let $Y_{ik}(t)$ indicate that the individual is under observation and eligible for the $k$th event at time $t$. A stratified PWP proportional-hazards model with a shared coefficient for covariate $X_i$ is $$ \lambda_{ik}(t \mid \mathcal{H}_{i}(t), X_i) = Y_{ik}(t)\lambda_{0k}(t)\exp(\beta X_i). $$ Here $\mathcal{H}_{i}(t)$ denotes the individual’s event history up to time $t$, $\lambda_{0k}(t)$ is the unspecified baseline hazard for event order $k$, and $\beta$ is the common covariate coefficient.

For two individuals in the same event-order stratum, with the same relevant history and one unit difference in $X$, the conditional hazard ratio is $\exp(\beta)$. For a binary covariate coded 0 and 1, this is the conditional hazard ratio comparing the group coded 1 with the group coded 0. The equality is across $k$: each order has the same hazard ratio, while $\lambda_{0k}(t)$ is allowed to vary with order.

For a vector of covariates $X_i$, the shared-effect version replaces $\beta X_i$ with $\boldsymbol{\beta}^{\mathsf T}X_i$, with the same coefficient vector in every event-order stratum. A shared coefficient applies on the log-hazard scale. It does not assert equal hazards, equal hazard differences, or equal recurrence counts across orders.

Assumptions Behind the Restriction

The model must be tied to a clear event-time scale and a clear event-order risk set. In a PWP total-time analysis, time is measured from the common origin; in a gap-time analysis, the relevant clock is time since the preceding event. The clock choice changes the hazard being modeled, so it must be specified before interpreting $\beta$. For the $k$th event, an individual cannot enter the risk set until the first $k-1$ events have occurred.

The shared-effect model relies on several assumptions. First, the event-order risk sets and event times are recorded correctly, and censoring is noninformative conditional on the modeled information: after conditioning on covariates and history, censoring must not selectively remove individuals with different unobserved future event hazards. Second, proportionality is assumed over the analysis time within each event-order stratum: the covariate’s hazard ratio does not change over time. Third, the covariate effect is assumed equal across the event orders included in the model. Finally, the chosen conditional hazard interpretation must match the scientific question.

These assumptions do not require the baseline hazard to be the same for all event orders, nor do they require an individual’s recurrent events to be independent. Dependence among events from the same individual affects uncertainty estimation; a subject-clustered robust variance may be used for inference when appropriate. Such a variance adjustment does not make a false shared-effect restriction true, repair incorrect risk sets, or change the coefficient’s estimand.

How the Shared Estimate Uses All Event Orders

With event-order-specific baseline hazards, the Cox partial likelihood contributions from the order-specific strata are multiplied, but they use the same $\beta$. Thus information from each order contributes to one constrained estimate. The shared estimate is not generally the unweighted average of separate order-specific estimates: the risk sets and information contributed by different event times can vary.

Justification: At an event time $t_{kj}$ in order-$k$ stratum, the partial-likelihood contribution for an event from individual $i$ is proportional to $$ \frac{\exp(\beta X_i)} {\sum_{r \in R_k(t_{kj})}\exp(\beta X_r)}, $$ where $R_k(t_{kj})$ is the risk set for that event order and time. Multiplying these contributions across all observed event times gives the partial likelihood for the shared coefficient. Its log-likelihood is a sum of stratum-specific log contributions, so its score is also a sum. The estimate solves the combined score equation, not a requirement that each stratum’s score separately equal zero. This is precisely how the common-effect constraint pools information while retaining separate baseline hazards.

Worked Example: Interpreting a Shared Coefficient

Suppose a hypothetical PWP total-time analysis models the first three event orders and estimates a common coefficient $\hat{\beta}=\log(1.50)=0.4055$ for a baseline treatment indicator coded 1 for treatment and 0 for control. Assume the risk sets are correctly constructed, censoring is conditionally noninformative, and the proportional-hazards and shared-effect assumptions are being used.

The estimated conditional hazard ratio at each modeled order is $$ \exp(\hat{\beta})=\exp(0.4055)=1.50. $$

The model therefore estimates a 50% higher instantaneous hazard of the $k$th event for treatment-coded individuals than for control-coded individuals, conditional on eligibility for that event order and the modeled history. It makes this same relative-effect claim for the first, second, and third events. It does not say that the absolute hazard is identical across orders: if the order-specific baseline hazards differ, the hazards being multiplied by 1.50 differ too.

Worked Example: A Shared Partial-Likelihood Calculation

Consider a small hypothetical calculation with one observed event in each of two event-order strata. In the first stratum, the risk set at the event time contains 2 exposed and 3 unexposed individuals, and an exposed individual has the event. In the second stratum, the risk set contains 1 exposed and 4 unexposed individuals, and an unexposed individual has the event. Assume one binary covariate $X$, a common coefficient $\beta$, and no ties at these event times.

Writing $z=\exp(\beta)$, the two partial-likelihood contributions are $z/(2z+3)$ and $4/(z+4)$. The combined partial likelihood is $$ L(\beta)=\frac{z}{2z+3}\frac{4}{z+4}. $$

The log-likelihood, apart from constants, is $\ell(\beta)=\log z-\log(2z+3)-\log(z+4)+\log 4$. Since $dz/d\beta=z$, its score is $$ \frac{d\ell}{d\beta} =1-\frac{2z}{2z+3}-\frac{z}{z+4}. $$

Setting the score to zero gives $$ 1=\frac{2z}{2z+3}+\frac{z}{z+4} =\frac{4z^2+11z}{2z^2+11z+12}. $$

Equating numerator and denominator yields $2z^2+11z+12=4z^2+11z$, so $z^2=6$. Because $z=\exp(\beta)>0$, the estimate is $\hat{z}=\sqrt{6}=2.449$, and $\hat{\beta}=\log(\sqrt{6})=0.8959$. Under the imposed shared-effect model, the estimated conditional hazard ratio is 2.449 at both event orders. This small calculation illustrates the pooled score equation; a real analysis uses all observed event times and the corresponding risk sets.

What Equality Across Orders Assumes and Implies

Equality means that, conditional on the model’s history and risk set, a one-unit increase in the covariate has the same multiplicative effect on the event hazard regardless of whether the event is first, second, or later. Equivalently, the model constrains the event-order-specific log hazard ratios to be equal. It does not mean that the consequences of an exposure are identical in every practical sense: later-event risk sets contain selected individuals who have experienced earlier events, and their histories may differ from those in earlier risk sets.

A common coefficient can be useful when the scientific question concerns a stable relative effect across recurrences and when a shared parameter provides a reasonable summary. It also reduces the number of covariate parameters, which may help when later event orders have few events. But the apparent precision gained by pooling is valid only if the restriction is credible. If effects truly differ by event order, the shared estimate is a compromise shaped by the risk sets and event information; it may obscure meaningful variation.

Worked Example: Same Hazard Ratio, Different Absolute Changes

Suppose, hypothetically, that the shared binary exposure effect is $\exp(\beta)=2$, while the baseline hazard at a particular time is 0.04 events per person-year for the first event and 0.20 events per person-year for the second event. Assume the proportional-hazards model applies at that time and compare otherwise equivalent individuals within each order’s risk set.

For the first event, the exposed hazard is $0.04\times2=0.08$ events per person-year, an absolute difference of $0.08-0.04=0.04$ events per person-year. For the second event, the exposed hazard is $0.20\times2=0.40$ events per person-year, an absolute difference of $0.40-0.20=0.20$ events per person-year.

The relative effect is 2 at both orders, but the absolute hazard differences are not equal. Neither calculation alone gives the expected total number of recurrences over follow-up; that requires accounting for changing event histories, eligibility, censoring, and the hazard over time.

Alternatives, Diagnostics, and Limits

The main alternative is to allow event-order-specific covariate coefficients rather than constrain them to be equal. That answers a different question and generally requires more parameters and more information, especially for later orders. Another choice is a marginal recurrent-event rate model, in which the coefficient describes a rate ratio rather than the conditional event-hazard ratio in a PWP model. These are not interchangeable interpretations; choose the model according to the target estimand and risk-set structure.

Before interpreting a shared estimate, check that event order, eligibility, follow-up time, and censoring are encoded as intended. Examine whether there are enough events and adequate covariate overlap in every included stratum; sparse or empty risk sets can make estimates unstable or impossible. Confirm that the stated clock matches the analysis, and that the covariate is measured and coded consistently. A common coefficient also presumes the covariate is modeled on the specified scale: changing a continuous covariate by one unit has effect $\exp(\beta)$, so units matter.

A fitted common coefficient cannot itself establish that the effects really are equal across orders. Equality is a model restriction. Comparing with a model that relaxes that restriction is one way to investigate it, but such a comparison needs enough data and appropriate inference. Detailed procedures for estimating event-specific effects and diagnosing event-order heterogeneity answer those separate questions. In all cases, a robust standard error addresses within-individual dependence in inference; it is not a test of the shared-effect assumption.

Common Pitfall

Wrong reasoning: “The coefficient is shared, so the hazard and treatment effect are the same for every recurrence.”
Correction: The model sets the conditional hazard ratio equal across orders. The baseline hazards, event-order risk sets, and absolute hazard differences may all vary. The equality is an assumption about relative effects on the model’s scale, not a claim that event processes are identical.

Key takeaway: A shared covariate effect in a stratified recurrent-event model is a common log-relative effect across event orders. State the conditional estimand and time scale, retain distinct order-specific baseline hazards when appropriate, and treat equality of effects as an assumption to assess rather than a fact guaranteed by pooling.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a stratified PWP model with a common coefficient $\beta$, what is constrained to be equal across event orders, and what may differ?
If $\hat{\beta}=\log(0.80)$ for a binary exposure coded 1 versus 0, give the shared conditional hazard ratio and interpret it.
Why is a shared-effect estimate not generally the simple average of separate estimates from each event order?
Does a subject-clustered robust variance make the shared-effect assumption valid? Explain.

Solutions. (1) The conditional log hazard ratio, and hence the conditional hazard ratio, is constrained to be equal; the baseline hazards and risk sets may differ by order. (2) The hazard ratio is $\exp(\hat{\beta})=0.80$, an estimated 20% lower conditional hazard for the group coded 1 than for the group coded 0 at each modeled order, under the model assumptions. (3) The shared estimate solves a combined score equation whose contributions depend on each order’s event times and risk sets, so their information need not be equal. (4) No. A robust variance can adjust uncertainty for within-individual dependence, but it cannot validate a false equality restriction or correct a misspecified model.